Guoce Xin

Exact Ehrhart Series of Birkhoff Polytopes via Constant Terms and Finite-Field Evaluation

Xinru Jiang, Guoce Xin, Chen Zhang +1

The Ehrhart series of the $n$ th Birkhoff polytope is $\sum_{r \geq 0} H_{n} (r) z^{r}$ , where $H_{n} (r)$ counts nonnegative integer $n \times n$ matrices whose row and column sums all equal $r…

A polynomial time algorithm for almost bounded denumerant

Guoce Xin, Chen Zhang, Zihao Zhang

Sylvester's denumerant $d (t; A)$ counts the number of nonnegative integer solutions to $\sum_{i = 1}^{N} a_{i} x_{i} = t$ , where $A = (a_{1}, \dots, a_{N})$ is a se…

Counterexamples to the Minimum Period Conjecture for Restricted Partition Functions

Feihu Liu, Jinlong Tang, Guoce Xin +1

For a finite sequence of positive integers $a = (a_{1}, \dots, a_{n})$ , the restricted partition function $q_{a} (k)$ denote the number of nonnegative integer solu…

Root-of-unity weighted trigonometric power sums: a constant term approach

Jiahang Liu, Guoce Xin

We present a unified method for evaluating finite sums of powers of cotangent, tangent, cosecant and secant weighted by primitive $k$ th roots of unity. The approach relies on const…

Ehrhart Theory of the Join of Two Lattice Polytopes

Feihu Liu, Sihao Tao, Guoce Xin

Inspired by research on the Cartesian product of two lattice polytopes, this paper investigates the Ehrhart theory of the join of two lattice polytopes. This is also a well-known o…

Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern

Feihu Liu, Sihao Tao, Guoce Xin

In Ehrhart theory, the well-known sign pattern problem asks: given a positive integer $d \geq 3$ and integers $1 \leq i_{1} < \dots < i_{k} \leq d - 2$ , does there exist a $d$ -dimensiona…

Unimodular Equivalence of Integral Simplices

Feihu Liu, Sihao Tao, Guoce Xin

Testing the unimodular equivalence of two full-dimensional integral simplices can be reduced to testing unimodular permutation (UP) equivalence of two nonsingular matrices. We cond…

Proof of a Conjecture on Young Tableaux with Walls

Zhicong Lin, Feihu Liu, Jiahang Liu +2

Banderier, Marchal, and Wallner considered Young tableaux with walls, which are similar to standard Young tableaux, except that local decreases are allowed at some walls. In this w…

MacMahon's $Ω_{\geq}$ operator: A computational framework

Feihu Liu, Guoce Xin

MacMahon introduced partition analysis in his book ``Combinatory Analysis'' as a computational technique for solving problems related to systems of linear Diophantine equations and…

The Sign Pattern Problem for Ehrhart Polynomials

Feihu Liu, Sihao Tao, Guoce Xin

We investigate the sign patterns of coefficients in the Ehrhart polynomial of the Cartesian product between the $r$ -th pyramid over the Reeve tetrahedron and the hypercube $[0, n]^…

Enumeration of Corona for Lozenge Tilings

Craig Knecht, Feihu Liu, Guoce Xin

Knecht considers the enumeration of coronas. This is a counting problem for two specific types of lozenge tilings. Their exact closed formulas are conjectured in [A380346] and [A38…

An $O (n lo g^{2} n)$ Algorithm for Computing Hankel Determinants up to Order $n$

Feihu Liu, Guoce Xin, Zihao Zhang

Given the rational power series $h (x) = \sum_{i \geq 0} h_{i} x^{i} \in C [[x]]$ , the Hankel determinant of order $n$ is defined as $H_n(h(x)) := \det (h_{i+j})_{1 \leq i,j \le…

Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS

Feihu Liu, Guoce Xin, Chen Zhang

In this paper, we provide an overview of Ehrhart polynomials associated with order polytopes of finite posets, a concept first introduced by Stanley. We focus on their combinatoria…

Order Polytopes of Dimension $\leq 13$ are Ehrhart Positive

Feihu Liu, Guoce Xin, Zihao Zhang

The order polytopes arising from the finite poset were first introduced and studied by Stanley. For any positive integer $d \geq 14$ , Liu and Tsuchiya proved that there exists a non…

On $P$ -partitions Extended by Two-Rowed Plane Partitions

Jingxuan Li, Feihu Liu, Guoce Xin

Inspired by Gansner's elegant $k$ -trace generating function for rectangular plane partitions, we introduce two novel operators, $φ_{z}$ and $ψ_{z}$ , along with their combinatorial…

A variation of the Morris constant term

Guoce Xin, Chen Zhang

The Morris constant term identity is important due to its equivalence with the well-known Selberg integral. We find a variation of the Morris constant term, denoted $h_{n} (t)$ , in th…

A polynomial time algorithm for Sylvester waves when entries are bounded

Guoce Xin, Chen Zhang

The Sylvester's denumerant $ d(t; \boldsymbol{a}) $ is a quantity that counts the number of nonnegative integer solutions to the equation $ \sum_{i=1}^{N} a_i x_i = t $, where…

A Fast Algorithm for Denumerants with Three Variables

Feihu Liu, Guoce Xin

Let $a, b, c$ be distinct positive integers such that $a < b < c$ and $g cd (a, b, c) = 1$ . For any non-negative integer $n$ , the denumerant function $d (n; a, b, c)$ denotes the number of soluti…

math.CO2024★ 3 cited

Three Simple Reduction Formulas for the Denumerant Functions

Feihu Liu, Guoce Xin, Chen Zhang

Let $A$ be a nonempty set of positive integers. The restricted partition function $p_{A} (n)$ denotes the number of partitions of $n$ with parts in $A$ . When the elements in $A$ are p…